USING DERIVATIVE INFORMATION ASSIGNMENT #7

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1991 AB 5 5. Let f be a function that is even and continuous on the closed interval [-3, 3]. The function f and its derivatives have the properties indicated in the table below.

x 0 0 1x< < 1 1 2x< < 2 2 3x< < ( )f x 1 Positive 0 Negative -1 Negative ( )f xʹ Undefined Negative 0 Negative Undefined Positive ( )f xʹ́ Undefined Positive 0 Negative Undefined Negative

(a) Find the x-coordinate of each point at which f attains an absolute maximum value or an absolute minimum value. For each x-coordinate you give, state whether f attains an absolute maximum or an absolute minimum. (b) Find the x-coordinate of each point of inflection on the graph of f. Justify your answer. (c) Sketch the graph of a function with all the given characteristics of f. 1989 AB 5

Note: This is the graph of the derivative of f, not the graph of f 5. The figure above shows the graph of f ʹ , the derivative of a function f. The domain of f is the set of all real numbers x such that 10 10x− ≤ ≤ . (a) For what values of x does the graph of f have a horizontal tangent? (b) For what values of x in the interval (-10, 10) does f have a relative maximum? Justify your answer. (c) For what values of x is the graph of f concave downward? Justify your answer.

USING DERIVATIVE INFORMATION ASSIGNMENT #7

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1997 AB 5 No Calculator 5. The graph of the function f consists of a semicircle and two line segments as shown above. Let g be the function given by

0( ) ( )

xg x f t dt= ∫ .

(a) Find g(3). (b) Find all values of x on the open interval (-2, 5) at which g has a relative maximum. Justify your answer. (c) Write an equation for the line tangent to the graph of g at x = 3. (d) Find the x-coordinate of each point of inflection of the graph of g on the open interval (-2, 5). Justify your answer. 2004 AB 4 (Form B)

4. The figure below shows the graph of f ʹ , the derivative of the function f, on the closed interval 1 5x− ≤ ≤ . The graph of f ʹ has horizontal tangent lines at x = 1 and x = 3. The function f is twice differentiable with f (2) = 6. (a) Find the x –coordinate of each of the points of inflection of the graph of f. Give a reason for your answer. (b) At what value of x does f attain its absolute minimum value on the closed interval 1 5x− ≤ ≤ ? At what value of x does f attain its absolute maximum value on the closed interval 1 5x− ≤ ≤ ? Justify your answers. (c) Let g be the function defined by ( ) ( ).g x xf x= Find an equation for the line tangent to the graph of g at x = 2. 1999 AB 5

5. The graph of the function f, consisting of three line segments, is given above. Let

1( ) ( )

xg x f t dt= ∫ .

(a) Compute g(4) and g(-2). (b) Find the instantaneous rate of change of g, with respect to x at x = 1. (c) Find the absolute minimum value of g on the closed interval [-2, 4]. Justify your answer. (d) The second derivative of g is not defined at x = 1 and x = 2. How many of these values are x-coordinates of points of inflection of the graph of g? Justify your answer.